Let f and g be two distinct normalized primitive cusp forms of even integral weights for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\) , respectively. In this paper, we are interested in the average behavior of coefficients \(\lambda _{f\otimes f\otimes \cdots \otimes _{l_{1}}f}(n)\lambda _{g\otimes g\otimes \cdots \otimes _{l_{2}}g}(n)\) associated to general product L-functions, where \(f\otimes f\otimes \cdots \otimes _{l_{1}}f\) and \(g\otimes g\otimes \cdots \otimes _{l_{2}}g\) denotes the \(l_{1}\) -fold and \(l_{2}\) -fold products of f and g, respectively. As an application, we also provide quantitative results concerning the sign changes of the sequence \(\{\lambda _{f\otimes f\otimes \cdots \otimes _{l_{1}}f}(n)\lambda _{g\otimes g\otimes \cdots \otimes _{l_{2}}g}(n)\}_{n-\text {squarefree}}\) in short intervals for certain ranges of \(l_{1}\) and \(l_{2}\) .
